It is a relatively simple exercise to prove that a well-ordered set is order-isomorphic to a subset of $\mathbb R$ (under the usual ordering) if and only if it is countable. You can say that $\mathbb R$ is "too small" to contain any uncountable well-ordered sets.
So my question is, can you embed bigger well-ordered sets in the long line? For those who don't know, the long line can be constructed by taking the minimal uncountable well-ordered set (i.e. $\omega_1$) and taking its Cartesian product with $[0,1)$ under the dictionary order. So obviously $\omega_1$ itself is emebeddable in the long line, just by taking the left endpoints of all the intervals $[0,1)$. But can you embed bigger uncountable ordinals, and if so how big? I'm guessing that you may be able to embed all well-ordered sets with cardinality less than or equal to $\aleph_1$, the cardinality of the set of countable ordinals.